Dimension-like functions of the type dim and universality

In [S.D. Iliadis, Universal Spaces and Mappings, North-Holland Math. Stud., vol. 198, Elsevier Science B.V., Amsterdam, 2005, xvi+559 pp.] base dimension-like functions of the type ind, Ind, and dim were defined. The domains of all these functions are classes of pairs (B,X), where B is a base of the space X. Also, positional base dimension-like functions of the type ind were defined whose domains are triples (Q,B,X), where Q is a subset of a space X and B is a base for Q in X. In [D.N. Georgiou, S.D. Iliadis, A.C. Megaritis, Dimension-like functions and universality, Topology Appl. 155 (2008) 2196–2201] and [D.N. Georgiou, S.D. Iliadis, A.C. Megaritis, The universality property for some dimension-like functions, Questions Answers Gen. Topology, in press] some dimension-like functions and positional dimension-like functions of type Ind, respectively are defined. All the above dimension-like functions are studied with respect to the property of universality. Here, we give positional base dimension-like functions of the type dim and prove the property of universality for these functions.